Applications cannot execute code while not in the foreground1 (ie. once you hit start, search or camera)

If applications take too long while exiting or being frozen, the process will be terminated

Frozen/dorment applications will be terminated if the system (or a foreground app) needs the memory

So while you can't terminate the process per se, it's not really necessary.

1 With the exception of background agents, which may run for 25 seconds every 30 minutes. Badly behaving background tasks (running too long, throwing errors) are automatically unscheduled after 2 strikes.

I guess if the HW buttons aren't working I just have to take out the battery.
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LouisMay 2 '12 at 14:35

If hitting the start button doesn't return you to the start screen within 10 seconds, it sounds like the phone has locked up. If it continues doing so, send it in to be fixed, as it is indicative of a driver or hardware fault.
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Richard SzalayMay 2 '12 at 23:55

You still need for super buggy apps like Audible. It doesn't throw errors but simple keeps drawing blank tabs and the only respite is to reboot the phone to kill the app. Of all OSes in the world, Windows (and its babies) is the one that needs taskkill the most!
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MrchiefSep 5 '13 at 20:50

Long press back again. The previous most recent app should be removed from the list.

You can keep doing this until there are no more recent apps in the list.

I don't know if this does anything more in terms of memory use or force quitting than just pressing the back button or start button, but it does bring some peace of mind that the app is probably not still running.

I tried the double tap tip, but for me it goes back to the home screen, then automatically goes "back" one app.
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LouisMay 4 '12 at 5:18

This only works if the foreground application is completely locked up. If it's not, the second back in the double tap is handled by the application and it simply returns to the previous page in the app.
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Richard SzalayMay 5 '12 at 2:59

@RichardSzalay I do not see the behavior you describe. Apps do not seem to handle the second back.
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jimuellerMay 7 '12 at 1:44

Wow! Long tap, double tap, then when the app appears in the window, long tap again and that app closes. Lovely. I didn't like not knowing how to close hose apps. Thank you!
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user858Dec 14 '12 at 3:04

You can press the windows/search/camera button to jump out of the application that is acting up. This will take you to the home/search/camera screen and the application that was running/acting up is made dormant.

Keeping the back button long pressed will show you all the dormant applications. If the system needs additional resources or if the program isn't reactivated after some time then the program is killed and it's resources are reclaimed.

If your application is acting up, you could also restart your phone to clear all dormant applications.

The only way to truly exit an application is to keep pressing the back button till you're out of all the application windows.

Or, if the problem is that you are just too deep in the navigation stack and you would have to press the back button a million times to quit it, you can just relaunch the app - press start button and launch your app (from start screen or from app list). This will start a new instance of the application with empty navigation stack and old instance of application (with big navigation stack) is thrown away.
From there is just one back-button-press to quit it.

From the new Windows 8.1 update, you can kill a process or an app by long pressing the back button. After a long back button press, tap the circled (x) button at the top right corner of the app you want to close as mentioned by @Tyler James Young.

Another method is, after a long back button press, you can kill/close any app by just swiping down that app!